Astronomical Images : Proportional parts of the Moon

Erasmus Reinhold

Astronomical Images

This Parisian edition was copied from the first edition of the commentary of Peuerbach by Erasmus Reinhold, printed in Wittenberg by Hans Lufft in 1542. Subsequently, in 1556, Charles Perier published a new edition, copied from the revised edition printed by Lufft in 1553, which contained additions to the theory of the Sun ([Theoricae] auctae novis scholiis in theoria Solis ab ipso autore). This woodcut is an enlarged and much improved version of the traditional figure of the proportional minutes of the Moon. It is related to a section of the treatise that examines the consequences of the eccentricity of the Moon. It concerns a device for the calculation of the equation of the argument of the Moon (aequatio argumenti Lunae), defined as the arc of the zodiac lying between the mean and true longitudes of the Moon, and dependent on the true argument of the Moon (argumentum Lunae verum), measured on the circumference of the epicycle, as it 'extends from the true apogee of the epicycle up to the centre of the body of the Moon'. As the distance of the centre of the epicycle from the centre of the World varies, the diameter of this epicycle, measured on the zodiac from the centre of the World, also varies. Therefore, to the same value of the true argument (measured on the epicycle) correspond unequal arcs of equations (measured on the zodiac): they are smaller near the apogee of the eccentric (and minimal at the apogee), larger near the perigee (and maximal at the perigee). The epicycle is called by Peuerbach 'the small circle' (circulus brevis), and the variation of the equations, for the same given argument, 'the variation of the diameter of the small circle' (diversitas diametri circuli brevis), abbreviated as 'variation, or diversity, of the diameter' (diversitas diametri). Reinhold's diagram has a legend. The outermost circle drawn around the centre of the World (T) probably represents the ecliptic, though it bears no symbol of the Signs, for the motion or longitude of the Moon is measured on it. HIKL, in the middle of the blackened orb, is the eccentric circle, which bears the centre of the epicycle. Its centre is S. H is the apogee (apogion) of this eccentric, and K the perigee (perigion). As in the preceding diagram (fol. 35r), D is the true apogee of the epicycle; F indicates the place of the Moon, in different positions of the epicycle, but always with the same true argument: according to the legend, all the arcs labelled DF are 'equal arcs of the epicycle, or equal true arguments' (DF arcus epicyclorum pares, seu argumenta vera paria). The line of the mean motion of the Moon is TB; the line of the true motion, TC; the equation of the argument, BC (for definitions of these lines and arc, see Reinhold, fol. 35r). We see that BC is minimal (omnium minimus) when the centre of the epicycle is at H, the apogee of the eccentric, and maximal, when this centre is at K, the perigee. The diversitas diametri is arc AC: it is null at the apogee and maximal at the perigee. TH is the line of the apogee (linea augis), TK the line of the opposite of the apogee (linea oppositi). 'The difference between them is equal to line STV, twice the value of the eccentricity, ST' (differentia utriusque, aequalis lineae STV, quae est duplum eccentricitatis ST). This difference is divided into sixty equal parts. In the diagram, this is shown by the circles numbered from 10 to 60 (talis differentia seu excessus â?¦ divisus est in 60 particulas aequales, ut patet in schemate adjectis numeris). These sixty equal parts are the 'proportional minutes'. These circles are concentric to the World. The innermost circle passes through K, the perigee of the deferent, and the outermost through H, the apogee. The space between these two circles is divided into six orbs by five concentric equidistant circles. We see on the diagram that when the centre of the epicycle is at the perigee, it is on line 60, and on line 0 at the apogee. The successive intersections of the eccentric deferent with the other lines mark the places where the centre of the epicycle is at 10, 20, 30, 40 and 50 proportional minutes. In Peuerbach's words, 'the equations of the arguments that are written in the tables are those that come about when the centre of the epicycle is in the apogee of the deferent' (Aequationes autem argumentorum, quae scriptae sunt in tabulis, sunt, quae contingunt, dum centrum epicycli in auge deferentis fuerit). The tables do not give the values of the equation when the centre of the epicycle is in other places, but it is possible to reckon them. If we know the centrum Lunae (the angle formed, at the centre of the World, by the lines drawn respectively to the apogee and to the centre of the epicycle), the tables indicate the proportional parts; and if we know the true argument, they indicate the value of the diversitas diametri when the centre of the epicycle is at the perigee of the eccentric (that is when the proportional minutes are 60); then we calculate the diversitas diametri corresponding to the exact position of the centre of the epicycle, and this value is added to the equation of the argument taken from the tables. For example, when the centre of the epicycle is at I, which corresponds to 15 proportional minutes (eastward), the centrum Lunae, HTI, being 75 degrees, and the true argument of the Moon, DF, 93 degrees, measured westward on the circumference of the epicycle (argumentum Lunae arcus epicycle DF contra seriem ut a sinistra versus dextram, sitque 3 Sig. 13 Grad.). For this value of the argument, the tables indicate that the equation is 4 degrees 53 minutes (when the centre of the epicycle is at the apogee). They also indicate that, for the same value of the argument, the diversitas diametri is 2 degrees 40 minutes when the centre of the epicycle is at the perigee, the place where the proportional minutes are 60. When the centre of the epicycle is at 15 proportional minutes, one quarter of 60, the diversitas diametri is only 40 minutes (one quarter of 2 degrees 40 minutes). This value of the diversitas diametri is added to the equation of the argument given in the tables (4 degrees 53 minutes), and we find that when the centrum Lunae is 75 degrees and the true argument 93 degrees, the equation of the argument is 5 degrees 33 minutes. Translated quotations of Peuerbach's Theoricae are from Aiton (1987). Quotations from Reinhold's commentary are translated or paraphrased by Isabelle Pantin.